Von Mises stress formula: 2D, 3D and worked example

1. Definition and direct answer

The Von Mises criterion, also called the distortion-energy or J₂ criterion, converts a stress tensor into one positive equivalent stress. It is mainly used to assess the onset of yielding in ductile materials.

The value is independent of coordinate orientation, but all components entered in one formula must be consistent and evaluated at the same point.

2. General 3D formula in Cartesian components

This expression uses the three normal stresses σx, σy, σz and the three independent shear stresses τxy, τyz, τzx. Every quantity must use the same unit, such as MPa.

Normal-stress signs matter in their differences. A shear-stress sign does not directly change the result because each shear term is squared.

3. Plane-stress 2D formula

For a thin plate loaded in its plane, σz = τyz = τzx = 0 is generally assumed. The general formula then reduces to this 2D expression.

One normal stress combined with shear is the special case σy = 0: σᵥ = √(σx² + 3τxy²).

4. Formula from principal stresses

In the principal coordinate system, shear stresses are zero. This form is convenient when σ₁, σ₂ and σ₃ come from an analytical calculation or finite-element software.

Pure hydrostatic stress, with σ₁ = σ₂ = σ₃, gives σᵥ = 0 because it produces no distortion.

5. Axial–bending–torsion case

At the same section point, first add axial and bending stresses with their signs. Torsion then provides shear stress τ.

Check opposite fibres when bending changes sign: the critical fibre is not always the one chosen intuitively.

6. Quick sanity checks

Three simple limiting cases help detect a formula, sign or unit error.

• If all shear stresses are zero and only one normal stress remains, σᵥ = |σ|.
• If only one shear stress τ remains, σᵥ = √3·|τ|.
• If σ₁ = σ₂ = σ₃, σᵥ = 0.
• The result must not change when every component is expressed in another coherent unit.

7. Limits of use

Comparison with yield strength assumes, among other things, a ductile material, a correctly evaluated stress state and an appropriate design rule. Brittle materials may require another criterion.

Fatigue, welds, plasticity, temperature, stress concentrations, stability and code requirements must be checked separately when relevant.